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Evaluating Numerical Expressions with Examples

ACTIVITY: Solve & Discuss It!  

An airline company charges additional fees for bags that do not meet the weight and size limits. For one flight, fees were charged for a total of 50 bags that were over the weight limit and 6 oversized bags. Find the total amount of fees collected for that flight. 

Solution: 

Step1: 

Total over-weight bags = 50 

Overweight fee for each bag=$49 

Fees collected for 50 over-weight bags = 50 x 49 =$2450 

Step2: 

Total oversized bags = 6 

Oversize fee for each bag = $75 

Fees collected for 6 oversize bags =6 x 75 =$ 450 

Step3: 

The total amount of fee collected for the flight = $2450 + $450 

∴ The total amount of fee collected for the flight =$ 2900 

Look for relationships you can use the order of operations to evaluate numerical expressions. 

The total amount of collected for the flight = (50 x 49 ) + (6 x 75 ) 

 =2450 + 450 

 =$2900 

  1. First did multiplication operation two times and  
  1. Then did addition operation. 

Order Of Operations 

  1. Evaluate parentheses and brackets from inside out. 
  1. Evaluate powers. 
  1. Multiply and divide from left to right. 
  1. Add and subtract from left to right. 

A Numerical Expression is a math expression that contains numbers and at least one operation.  

3 + 4, (5)(³∕₄), and 8 ÷2 + 0.5 are numerical expressions. 

Essential Question  

How do you write and evaluate numerical expressions? 

Some expressions look difficult because they include parentheses and brackets.  

You can think of brackets as “outside” parentheses.  

Evaluate the following numerical expression. 

½ x 4² -[2 + (3.6 ÷ 0.9)] 

Use the Order of Operations to Evaluate Numerical Expressions 

Example 1 

Step (1) Evaluate parentheses and brackets from inside out. 

½ x 4² – [2 + (3.6 +0.9)] = ½ x 4² – [2 + (3.6 +0.9)] ………Evaluate inside the parentheses. 

                                             =  ½ x 4² – (2+4) ……………. Evaluate inside the brackets. 

                                             =  ½ x 4² – 6 

Step(2) Evaluate powers. 

                                            =   ½ x 4² – 6 

                                            =    ½ x 16 – 6 

Step (3) Multiply and divide from left to right.  

                                              =   ½ x 16 – 6              

                                              = ½ x 16/1 – 6            [Multiply or divide 2 from left to right] 

                                              = 8 – 6 

Step (4) Add and subtract from left to right.             

                                              = 8 – 6 [Add or subtract from left to right]  

∴ ½ x 4² – [2 + (3.6 +0.9)] = 2. 

Try It!  

Evaluate the numerical expression at the right. Convince Me! Why is it important to follow the order of operations? 

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Example 2: 

Evaluate the numerical expression  

The value of the numerical expression is 90.25. 

Example 3 

Insert grouping symbols in the expression so that it has a value of 10.   

40 – 7 + 33 x ¾

Solution: 

Case 1 : Place the parentheses around 40 – 7 and evaluate. 

∴ 57 ¾ ≠ 10, so place the grouping symbols in another location.  

Case 2 : 40 – 7 + 33 x ¾

Place the parentheses around 7 + 33 and evaluate 

Inserting grouping symbols around 7 + 33 gives numerical expression a value of 10 

Try it: 

A: Evaluate the numerical expression: 

3.2² – [(9 x 4) + 9] x (⅓)² 

B: Insert the grouping symbols so that the numerical expression has a value of 80. 

              6 + 12 x (⅔)² x 3 + 7 

Answers: 

A. 3.22 – [(9 x 4) + 9] x (⅓)² = 10.24 – [(36) + 9] x ¹∕₉

                                                  =10.24 – [ 45] x ¹∕₉

                                                  = 10.24 – 45 x ¹∕₉

                                                  =10.24 – 5 

                                                  =5.24 

B. 6 + 12 x (⅔)² x 3 + 7  

Step 1: 

To insert grouping symbols so that the expression has a value of 80 use the trial-and-error method. Try putting parentheses in some place and calculating the value. If the result isn’t what you need, try putting the parentheses in some other place. Repeat until you find the correct result. 

Step 2: 

 6 + 12 x (⅔)² x 3 + 7 = (6 + 12) x (⅔)² x 3 + 7 

                                         = 18 x (⅔)² x 3 + 7 

                                         =18 x ⁴∕₉ x 3 + 7 

                                         =8 x 3 + 7  

                                         =24 + 7 

                                         =32  

Step 3: 

Second try 

6 + 12 x (⅔)² x 3 + 7 =  (6 + 12) x (⅔)² x (3 + 7) 

                                         =  18 x (⅔)² x 10  

                                         = 18 x ⁴∕₉ x 10 

                                         = 8 x 10 

                                         = 80 

The order of operations is a set of rules used to evaluate expressions that include more than one operation. 

Order Of Operations 

  1. Evaluate parentheses and brackets from inside out.  
  1. Evaluate powers.  
  1. Multiply and divide from left to right.  
  1. Add and subtract from left to right. 

Let’s check our knowledge 

  1. In the expression (21 – 3) x (7 + 2) ÷ (12 – 4), what operation should you perform last? Explain. 
  2. Evaluate each expression. 
    1. 5² +  (6.7 – 3.1)
    2. (8.2+5.3)÷5. 
    3. (1.5 −0.5²)÷[(3+2)x 2].
    4. 36.8 ÷ [11.5 – (2.5 x 3)]². 
  3. Insert grouping symbols so that the expression has the given value.
    12 x 3² + 36     Target value: 540. 
  4. Cory bought some baseball equipment. He used a coupon for off the price of the bat and glove. Write and evaluate a numerical expression to find the total cost of the bat, the glove, and 3 baseballs.
  5. Evan says that the value of numerical expression.  
    0.2²+12 + (1.5 X 4) is 32.04 Do you agree? 

Answers: 

  1. (21 – 3) x (7 + 2) ÷ (12 – 4) =18 x 9 ÷ 8  

                = 162÷8 

                = 20 ²∕₈

               = 20 ¹∕₄

Last operation division 

    1. 5²+(6.7−3.1)= 25 + 3.6 = 28.6 
    2. 8.2+5.3) ÷5 = 13.5 ÷5  
                        =2.7 
    3. (1.5 −0.5²)÷[(3+2) x 2] = (1.5 – 0.25) ÷ [5 x 2]  
                                          =1.25÷ 10  
                                          =0.125 
    4. 36.8÷ [11.5 – (2.5X3)]²=36.8÷[11.5−7.5]²
                                          =36.8÷4²
                                          =36.8÷16 
                                          =2.3 
  1. Step1: 12 x  3² + 36 =12 x 9+36 =108+36=144 
    Step2: 12 +3² x 36 =12+ 9 x 36 = 12+324=336 
    Step3:  12 x 36 +3²= 432 + 9 = 441
    Step4: 12 x (3²+36) = 12 x (9+36)  
                                   = 12 x 45  
                                   = 540 target reached….. 
  2. Given that, 
    Cost of bat = $69 
    Cost of glove = $75 
    Cost of ball = $5.50
    Cost of one bat, one glove and 3 gloves  
    = $69 + $75 + 3 x $5.50 
    = 69 + 75 + 16.5 
    =$160.5 
  3. I don’t agree with Evan, 
    Because, the solution for the given expression is 
    0.2²+12 + (1.5 X 4) = 0.04 + 12 + 6 
    = 18.04 
    Evam said that the answer is 32.04, but the answer is 18.04.  
    So, he is wrong.  

Exercise:

  1. Evaluate each expression.
  2. Insert grouping symbols so that the expression has the given value.Target value 23.
  3. Use the order of operations to evaluate.
  4. William bought 6 bananas for 50 cents each and 1 apple for 80 cents. Write the numerical expression to represent this situation and find the total cost in dollars?
  5. The price of the shirt is 200 dollars. The store manager gives a discount of 100 dollars. A man and his brother bought 4 shirts and then share the cost with his brother. Write a numerical expression to represent this situation, and then find the price paid by each brother?
  6. James pays 40 dollars for materials to make earrings. He makes 10 earrings and sells 7 for 10 dollars each and 3 for 5 dollars each. Write a numerical expression to represent this situation and then find James’s profit?
  7. Benjamin bought two burgers for $20 each and 3 medium French fries for $10 each. Write a numerical expression to represent this situation and find the cost?

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